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During Goblet of Fire, Voldemort lured Harry away from Hogwarts and into a Muggle cemetery. This was done during the Triwizard tournament to make Harry’s death look like an accident.
Harry was supposed to have a Trace on him, so why couldn’t the Ministry tell an underage wizard was performing magic (Expelliarmus) in a Muggle cemetery? Or (as the Trace seems to pick up ANY magic used near a young wizard/witch) magic used by the Death Eaters?
I don’t think this is answered in canon, but I can come up with two theories:
The Trace is only active during school holidays.
During term-time, students are always in Hogwarts (where magic is freely permitted) or Hogsmeade (a community that contains no Muggles), so there’s very little need for the Trace. There are very few, if any, instances of students leaving the school grounds and surrounding area during term-time.
If not, then the Ministry are getting a lot of spurious data from students. Either Hogwarts and Hogsmeade are specially enchanted to neutralise the Trace, or the Ministry just ignores anything they receive during term-time. However, I tend to reject the latter possibility, because if that were the case, then they’d be able to use information obtained with the Trace to know that dark magic was performed in the courtyard, and that Voldemort might have returned.
The graveyard was shielded against the Trace, or detection spells in general.
It’s plausible that you can enchant magical-only areas to be impervious to the Trace. Plenty of locations would merit this treatment – Hogwarts, Hogsmeade, Diagon Alley. If this magic exists, then it’s possible that Voldemort/Wormtail applied similar enchantments to the graveyard, thus preventing the Ministry from tracing them before Harry was dead.
In The Half-Blood Prince Chapter 26, Dumbledore says:
"I do not think you will count, Harry: You are underage and un-qualified. Voldemort would never have expected a sixteen-year-old to reach this place: I think it unlikely that your powers will register compared to mine."
Perhaps this works as a general rule, so in the graveyard, Voldemort + his Death Eaters' power levels masked Harry's, preventing the Ministry from detecting the Trace. On the other hand, the Trace was inconsistently mentioned...
alexwlchan's second theory mentioned on the HP wiki seems also plausible, since Voldemort wanted to stay hidden at the time and was not be the type of person to miss something involving human magic.
There are quite a few times when spells are cast around Harry in a Muggle area, but the Ministry doesn't seem to know or care. For example:
The common thread is that the Ministry seems to be aware of any adult wizards nearby when the Trace is activated. Given how many wizards have pre-school age children, the Trace should be activated thousands of times a day. I can't imagine they have full-time staff reviewing every single instance, and they likely have some automated system to weed those instances out. We can assume that when the infiltrators in the Ministry wanted to get Harry at the beginning of Deathly Hallows, they changed the process in some way so they'd be informed.
(As an out-of-universe explanation, when we learn of the Trace in Deathly Hallows, it always struck me as a convenient plot excuse to have a giant battle in the sky. “If we Apparate, the Ministry will know!”… And? By then, Harry will be safe in the Burrow. The fact that Harry somehow went through an entire underage magic trial without ever being told about the Trace, and the fact that it only has implications for the last book while contradicting much of the earlier books, it's just always seemed contrived to me.)
From a re-read of the chapters; there is no evidence that the Ministry was not alerted by the Trace.
Remember that almost everyone who is anyone is at the Triwizard Tournament. This includes:
Harry was gone, lets say about 30-45 minutes in total1. This includes,
It is safe to assume that Cedric's body was still warm when Harry got back. This doesn't leave much time for the Ministry to do much about it.
By the time Harry had escaped, it would not have taken long for the DEs and Voldemort to 'leg-it'.
From what we've seen and know about the Trace is that it knows two things for sure:
1. Magic has been used by, or around, an underage wizard.
2. The (approximate) location of the incident.
That's it. It doesn't know for sure sure who the underage wizard was2, or the nature of the magic (very powerful dark magic or a simple Lumos).
So in effect, even if the Ministry was alerted, what they know for sure is that somewhere near Little Hangleton Cemetery some magic was used by, or near, an underage wizard.
The spectators of the Triwizard tournament already know that something is wrong, and that Harry and Cedric appear to be missing, and the extra information from the Ministry would have been inconsequential at that point...
There is no way to know whether or not the Ministry was alerted, and even if they were, it would not have brought much help at that time3.
A theory: The Trace apparently detects Muggles around the spellcaster. It is possible that the whole trace was discarded because there were no Muggles around.
Yet another theory: Voldemort had his people in various parts of the Ministry, even so shortly after his return. Maybe the tracing information fell into hands of such a Voldemort's minion and was discarded.
Another theory (this time about the internals of the Trace): I think that it counts only with some sort of "registered" spellcasters. For example, Harry was not charged with tidying, making light etc. at the beginning of the Order of the Phoenix, because all of those spells were associated with someone else by the system. Conversely, Dobby probably does not fall into the "registered spellcasters" category, so his Wingardium Leviosa in the Chamber of Secrets was attributed to the nearest registered wizard, which was Harry.
I think the fact it happened during the Triwizard Tournament may have had something to do with it.
During this tournament he would be expected to cast spells and have spells cast at him, possibly even dark spells. The location would be off but the spells and action would be normal, perhaps they only investigate the location and further information from the trace if it is unexpected or illegal.
There is also the political issues to think of, Harry is competing with students from other countries. Having England use trace magic on their student while he is competing with students from other countries could be seen as a breaking of international treaty.
I have two theories:
The cemetery was not in a muggle area. The Gaunts' house was nearby, and magic must have been performed there regularly, at least while the Gaunts were alive. We're not told if any other magical families lived nearby. When Tom Riddle went to find his family and ended up killing his father, he was still at Hogwarts (HBPrince - when Tom is talking to Slughorn in his office he is already wearing the Gaunts ring). But his magic was presumably not detected, unless he was already so talented that he could evade the trace.
